Packing Dimension Research Articles

Column generation approach for 1.5-dimensional cutting stock problem with technical constraints

In this study, the 1.5-dimensional cutting stock problem with technical constraints is considered. In the literature, this problem is also defined as a strip packing or open dimension problem. When given a strip of infinite length and bounded width, the problem is to define a packing of rectangular objects into a strip that minimizes its final length. Technical constraints, such as the order type and the number of strips, are indispensable in real life; however, they are often neglected in the literature because they make the problem difficult to solve. Only one study was reached in the literature that took into account technical constraints, but in that mentioned study, only a mathematical model was proposed for the problem. In this context, our aim is to solve the problem with a more effective approach. The research question in this study is the usability of the column generation technique to solve the 1.5-dimensional cutting stock problem. In this study, the column generation approach was proposed for the first time for the considered problem. To demonstrate the performance of the proposed solution method, randomly generated test problems were solved with GAMS/Cplex. As we report the results, proposed column generation approach (CG) reaches very close (such as %1 and %2 error) solutions to integrated mathematical model (IM) for small sized problems in a second. On the other hand, while CG solved all the problems in a reasonable time, IM could not produce a feasible solution to some problems. Numerical experiments showed that the column generation algorithm outperforms the integrated mathematical model for the problem.

Lastarria volcano, a major emitter of boron and chalcophiles in northern Chile and the Central Volcanic Zone

This study examines the trace element chemistry of fumarolic deposits and fumarolic plumes at Lastarria volcano, located in northern Chile. Fumarolic deposits were manually collected, while fumarolic plumes were sampled using the filter pack technique; after sample preparation, both sample types were analyzed by ICP-MS. The research focuses on chalcophile elements (As, Cd, Sb, Te, Cu, Zn, Tl, Pb, and Bi) and B, with the findings revealing significant enrichment of As and Pb in fumarolic deposit samples, especially in medium- and high-temperature fumarolic vents, respectively. On the other hand, filter pack measurements in the fumarolic plume indicate a notable enrichment in Te, B, As, and Tl, with Sc-normalized enrichment factors within the range of 104–106. From these elements, B, Tl, and Te appear to be less scavenged at the vent, presenting more significant enrichments in fumarolic plumes than in fumarolic deposits. By comparing our results with subduction-related volcanoes such as Etna, Stromboli, Lascar, Masaya, or La Soufrière de Guadeloupe, Lastarria fumarolic plume can be regarded as the one with the highest concentrations of B and As ever recorded. Moreover, Lastarria has chalcophile enrichments comparable to those volcanoes with shallower magma chambers and high-temperature magmatic emissions such as Stromboli. The higher abundance of chalcophiles suggests a significant magmatic influence feeding Lastarria fumaroles and the presence of ligands, such as Cl, which can transport significant amounts of chalcophiles at shallower levels of the crust during late-stage magma crystallization. Large amounts of chalcophiles released by Lastarria and Lascar volcanoes could be responsible for the naturally elevated concentrations of these elements in the Altiplano-Puna area. Future studies should focus on the lifetime of chalcophiles in the Central Andean atmosphere, where these elements could reside for a longer time given the dominant hyperarid environment.

Bishop–Jones’ theorem and the ergodic limit set

Abstract For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

Spectral packing decomposition of CMV matrices through power-law subordinacy

This note gives a classification for the spectral measures of CMV matrices, which are special unitary five-diagonal matrices related to orthogonal polynomials on the unit circle, with respect to packing measures.

Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set of each CIFS is zero and the packing measure of the limit set with respect to the Hausdorff dimension is positive. In this paper, we show that the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal, and the proper-dimensional packing measure of the limit set is finite.

The packing measure of the linear Gauss system

For every k \in \mathbb{N} , let f_{k}:[\frac{1}{k+1}, \frac{1}{k}] \to [0,1] be decreasing, linear functions such that f_{k}(\frac{1}{k+1}) = 1 and f_{k}(\frac{1}{k}) = 0 , k = 1, 2, \dots . We define the iterated function system S_{n} by limiting the collection of functions f_{k} to first n, meaning S_{n} = \{f_{k} \}_{k=1}^{n} . Let J_{n} denote the limit set of S_{n} . Then, \lim\limits_{n\to \infty}\mathcal{P}_{h_n}(J_{n}) = 2 , where h_{n} is the packing dimension of J_{n} and \mathcal{P}_{h_n} is the corresponding packing measure.

General fractal dimensions of typical sets and measures

Consider (Y,ρ) as a complete metric space and S as the space of probability Borel measures on Y. Let dim‾BΨ,Φ(E) be the general upper box dimension of the set E⊂Y. We begin by proving that the general packing dimension of the typical compact set, in the sense of the Baire category, is at least inf⁡{dim‾BΨ,Φ(B(x,r))|x∈Y,r>0} where B(x,r) is the closed ball in Y with center at x and radii r>0. Next, we obtain some estimates of the general upper and lower box dimensions of typical measures in the sense of the Baire category. Finally, we demonstrate that if S is equipped with the weak topology and under some assumptions then the set of measures possessing the general upper and lower correlation dimension zero are residual. Furthermore, the general upper correlation dimension of typical measures (in the sense of the Baire category) is approximated through the general local lower and upper entropy dimensions of Y.

General fractal dimensions of graphs of products and sums of continuous functions and their decompositions

This study takes a broad approach to the fractal geometry problem and proposes an intrinsic definition of the general box dimensions and the general Hausdorff and packing dimensions by taking into account sums of the type ∑ih−1(sg(ri)) for some prescribed functions h, g and for all positive real s. Our primary aim is to conduct a more comprehensive exploration of the fractal dimensions exhibited by graphs resulting from the products and sums of two continuous functions. This article is entirely devoted to the investigation of this specific problem domain. In the course of our research, we furnish explicit formulas for computing both the general upper box dimension and the general lower box dimension associated with the graphs formed by the products and sums of two continuous functions. Additionally, we shed light on the decomposition of continuous functions into the sum of two continuous functions, employing the framework of the general lower and upper box dimensions. Furthermore, we establish an upper bound for the general upper box dimension applicable to each element within a finite ring of polynomials derived from continuous functions defined over the real number field R. Our inquiry also extends to the examination of general Hausdorff and packing dimensions for the graphs generated through both the summation and multiplication of two continuous functions.

Exceptional projections of sets exhibiting almost dimension conservation

We establish a packing dimension estimate on the exceptional sets of orthogonal projections of sets satisfying an almost dimension conservation law. In particular, the main result applies to homogeneous sets and to certain graph-directed sets. Connections are drawn to results of Rams and Orponen.

New fractal dimensions of measures and decompositions of singularly continuous measures

By introducing a novel expression for the Hausdorff and packing measures with respect to gauge functions, we establish a connection between different approaches to defining the exact Hausdorff and packing dimensions of measures. Furthermore, we define the upper and lower Hausdorff and packing dimensions of a Borel probability measure and demonstrate that these dimensions can be represented using an entropy formula. This paper also addresses an intriguing question raised multiple times regarding the structural representation of singular probability measures. We provide an affirmative response to this question by utilizing characteristic measures.

Note on the Multifractal Formalism of Covering Number on the Galton-Watson Tree

We consider, for t in the boundary of Galton-Watson tree (∂T), the covering number Nn(t) by cylinder of generation n. For a suitable set I and a sequence (sn,γ), we establish almost surely, and uniformly on γ, the Hausdorff and packing dimensions of the set {t ∈ ∂T : Nn(t) − nb ∼ sn,γ} for b ∈ I.

Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, and laws of the iterated logarithm (LILs). It is obtained that the set of fast points at which LILs fail in this process, and occur infinitely often, is a random fractal, the size of which is evaluated by its Hausdorff dimension. These points of this process are everywhere dense with the power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension dimP(E) of the target set E.

A variational principle in the parametric geometry of numbers

We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof of our variational principle relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. We use our variational principle to compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs have equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne.

Conditional aggregation operators defined by the Choquet integral and the Sugeno integral with respect to general fractal measures

Conditional aggregation operators are defined by the Choquet integral and the Sugeno integral with respect to a monotone set function that assesses positive measure of the conditioning set. General Hausdorff and packing measures are introduced and examples of infinite s-sets with positive and finite generalized Hausdorff and packing measures are constructed and their fractal dimensions are compared. Coherent upper conditional previsions on the linear space of all Choquet integrable random variables are defined by the Choquet integral with respect to the general Hausdorff and packing measures when the conditioning event has positive and finite generalized Hausdorff and packing measures in its respective fractal dimensions. Conditional aggregation operators are defined by the Sugeno integral with respect to general Hausdorff and packing measures on the class of all Sugeno integrable random variables. Actually, the general Hausdorff and packing dimensions are proven to be the Sugeno integral with respect to the Lebesgue measure of the general Hausdorff and packing measures respectively.

A generalized family of transcendental functions with one dimensional Julia sets

A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any s ∈ ( 0 , + ∞ ] , there is a function taken from this family with the order of growth s. Baker proved that the Hausdorff dimension of a transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bishop's construction, the family of functions here have arbitrarily positive or even infinite order of growth.

Fractals and the monadic second order theory of one successor

We show that if $X$ is virtually any classical fractal subset of $\mathbb{R}^n$, then $(\mathbb{R},,+,X)$ interprets the monadic second order theory of $(\mathbb{N},+1)$.This result is sharp in the sense that the standard model of the monadic second order theory of $(\mathbb{N},+1)$ is known to interpret $(\mathbb{R},,+,X)$ for various classical fractals $X$ including the middle-thirds Cantor set and the Sierpinski carpet.Let $X \subseteq \mathbb{R}^n$ be closed and nonempty.We show that if the $C^k$-smooth points of $X$ are not dense in $X$ for some $k \geq 1$, then $(\mathbb{R},,+,X)$ interprets the monadic second order theory of $(\mathbb{N},+1)$.The same conclusion holds if the packing dimension of $X$ is strictly greater than the topological dimension of $X$ and $X$ has no affine points.

Dirichlet is not just bad and singular in many rational IFS fractals

Abstract For $m\ge 2$, consider K the m-fold Cartesian product of the limit set of an iterated function system (IFS) of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and K does not degenerate to singleton, we construct vectors in K that lie within the ‘folklore set’ as defined by Beresnevich et al., meaning that they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within K; however, we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.

Dimensions of projected sets and measures on typical self-affine sets

Let T1,…,Tm be a family of d×d invertible real matrices with ‖Ti‖<1/2 for 1≤i≤m. For a=(a1,…,am)∈Rmd, let πa:Σ={1,…,m}N→Rd denote the coding map associated with the affine IFS {Tix+ai}i=1m. We show that for every Borel probability measure μ on Σ, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of π⁎aμ is constant for Lmd-a.e. a∈Rmd, where π⁎aμ stands for the push-forward of μ by πa. In particular, we give a necessary and sufficient condition on μ so that π⁎aμ is exact dimensional for Lmd-a.e. a∈Rmd. Moreover, for every analytic set E⊂Σ, each of the Hausdorff, packing, lower and upper box-counting dimensions of πa(E) is constant for Lmd-a.e. a∈Rmd. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.

Extending the reach of the point-to-set principle

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces Rn. These are classical questions, meaning that their statements do not involve computation or related aspects of logic.In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X. We first extend two algorithmic dimensions—computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim⁡(x) and Dim(x) to individual pointsx∈X—to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families.We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces.

Holomorphic motions, dimension, area and quasiconformal mappings

We describe the variation of the Minkowski, packing and Hausdorff dimensions of a set moving under a holomorphic motion, as well as the variation of its area. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles.